Chemical Systems and Equilibrium
A- Chemical systems
Reading Assignment: p. 1- 4; Chapter 4, p. 205.
A volume or space of the earth of distinct chemical and physical properties in which
we are interested.
Types of systems:
(1) Open system: An open system is one that is allowed to exchange energy and
mass with its environment.
(2) Closed system: Is a system that is allowed to exchange energy but not mass
with its environment.
(3) Isolated system: Is a system that is not allowed to exchange energy or mass
with its environment.
Phase: Is that part of the system with distinct chemical and physical properties, and
which can be separated from the system physically. The number of phases in a
system is denoted by "p".
System Components: Smallest number of chemical constituents needed to make up
all phases in the system (i.e. to define the system). The minimum # of components in
a system is denoted by "c".
Variance (also known as the degrees of freedom "f"): Is the maximum number of
intensive properties that can be changed independently within a system without
causing the appearance or disappearance of any phase in this system.
Properties of a system:
System properties or variables are either extensive or intensive. Extensive properties
are those that depend on the mass of the system as a whole. They include volume,
energy, mass, and entropy. Intensive properties are those that are independent of the
mass of the system, and include: T, P, and chemical potentials.
B- The Phase Rule
The phase rule was formulated by J. Willard Gibbs (1874) to determine the variance
or degrees of freedom of a system “f” of a known # of components “c” and phases
“p”. The variance of a system can be expressed as the total variables of this system
minus the "fixed variables". For any system of "c" components and containing "p"
phases, the total number of variables is c.p + 2, where the “2” represents the two
variables P and T. The fixed variables are given by the number of equations needed
to fully define the composition of the system, and expressed as: c(p-1) + p.
Accordingly, the degrees of freedom will be given by:
f = cp + 2 - [cp - c + p]
f=c-p+2
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The phase rule therefore allows us to determine the minimum number of variables
that must be fixed in order to perfectly define a particular condition of the system
from a knowledge of the number of system components and phases. Note that this
"number of variables" cannot be negative (i.e. f  0). The phase rule also allows us to
determine the maximum number of phases that can coexist stably in equilibrium; e.g.
if a system has 20 components, according to the phase rule, the maximum number of
phases in equilibrium will be 22 (when f = 0). Accordingly, if the number of phases
present exceeds that calculated by the phase rule (after the number of components
has been correctly identified), then not all these phases are in equilibrium!
Based on the phase rule, the condition of a system can be described as invariant,
univariant, divariant .....etc., if f = 0, 1, 2,.....etc. respectively, where:
(i) an invariant state is one in which neither P nor T nor X (composition) can be
changed without causing a change in the number of phases present in the system. On
the P-T diagram for the one-component system "SiO2" (which shows the P-T
stability fields of the silica polymorphs; Fig. 1), an invariant state is represented by a
point as "C" where three phases (cristobalite, high quartz and tridymite) coexist.
(ii) a univariant state is one in which either P or T need to be specified in order to
fully define the system. A univariant mineral assemblage can therefore be maintained
if a change in one variable (e.g. P) is accompanied by a dependent change in another
(e.g. T), but if one of these two variables is held constant while the other is changed,
the assemblage is no longer stable, and the system is no longer univariant. An
example of this is given by point "B" of Fig. 1 (or any point lying along any of the
curves defining the P-T limits of the different phases).
(iii) a divariant state is one in which two variables have to be specified in order to
fully define or characterize the system. An example of a divariant state is given by
point "A" (Fig. 1), or any point lying in the stability fields of one of the polymorphs.
The condensed Phase rule:
In cases where either P or T are held constant, one can apply the "condensed phase
rule" given by the formula:
f=c-p+1
This is simply because the total number of variables within the system has now
become pc+1, since only one of the two intensive properties of the system (P and T)
is allowed to vary. The condensed phase rule is quite helpful in understanding
isobaric T-X or isothermal P-X diagrams, and in experimental geochemistry, where
either P or T are held constant to investigate the dependence of the system on the
other intensive variable.
Application of Phase Rule to a system with dissolved species:
In cases where the system consists of dissolved species, defining the number of
components may become challenging. If we decide to consider every dissolved ionic
species a component, then the phase rule is best expressed by the formula:
f = c’ - p –r + 2
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where c’ is the number of different chemical species in the system, and r the number
of “auxiliary” restrictions necessary to fully define the system.
To fully appreciate this, we consider the example of a system containing the mineral
calcite at saturation and in equilibrium with gaseous CO2. The system contains 3
phases (water, calcite, and CO2), and may be considered to conatin 3 components as
well (as we attempt to minimize c in accordance with the definition). Application of
the phase rule f = c-p+2 would indicate that the system is bivariant (3-3+2).
However, calcite dissociates in water giving ionic species, water dissociates to H+
and OH-, whereas CO2 dissolves in water to form carbonic acid. Possible equilibria
in the system therefore include:
CO2 + H2O = H2CO3
H2CO3 = H+ + HCO3HCO3- = H+ + CO32H2O = H+ + OHCaCO3 = Ca2+ + CO32Once we decide to consider the ionic species as system components, c’ = 9 (Ca2+,
CO32-, HCO3-, H+, OH-, H2O, CO2, H2CO3, and CaCO3). “r” then becomes the
number of equations necessary to fully define the system (solve for some
concentrations knowing others). Because c = c’-r, r must be equal to 6. The six
equations necessary to fully define the system become the 5 reactions listed above
(one equation for each reaction, each expressing the equilibrium constant), and an
equation that defines the charge balance of the system (all waters have to be charge –
balanced):
2[Ca2+] + [H+] = [OH-] + [HCO3-] + 2[CO32-]
C- Phase Diagrams
Types of phase diagrams:
Phase diagrams are those which show the stability fields of, and relations between
different phases as a function of such variables as P, T, composition (X, a, or f), .....
etc. The most informative diagrams would be those in which perhaps as many
variables as possible are plotted against each other (e.g. P-T-X diagrams, Fig. 2), but
these would be difficult to construct or read. Instead, only two variables are plotted
against each other. Examples of such phase diagrams include: P-T diagrams, P-X
diagrams, T-X diagrams, plots of two compositional variables against each other
(activity – activity diagrams), and ternary diagrams where three compositional
variables are plotted at the apices of a triangle.
Because of the importance of temperature and composition in the crystallization of a
magma, T-X diagrams and ternary "projections" (usually constructed at constant P)
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are the ones most widely used in studying igneous systems. Because T-X diagrams
deal with two component systems (in most cases), they are often referred to as "phase
diagrams for binary systems". T-X diagrams in which a third component is
considered to be always present are known as pseudobinary diagrams. For
metamorphic systems where we assume that the rocks do not change their bulk rock
composition, P-T diagrams are the most widely used. Nevertheless, T-X, P-X, and
ternary diagrams are also very useful for illustrating the compositional changes that
take place in minerals as a function of T or P. For low T aqueous systems,
metasomatic rocks or hydrothermal systems, phase diagrams in which more than 2
compositional variables (including ionic species in some cases) are expressed as
ratios are very useful (e.g. Figs. 3 & 4), especially for studying the compositional
evolution of the fluids over time. Keep in mind that all these different types of phase
diagrams have to obey the phase rule!
Construction of phase diagrams:
Phase diagrams are constructed by carrying out experiments under specified
conditions of P, T and X. Minerals or compounds representing the system are mixed
with each other in a capsule made of non-reacting material (e.g. Au or Pt), sealed,
then placed in a furnace/ hydrothermal bomb/ piston cylinder, where they are allowed
to reach a specific P and T. After a sufficient time period, the experiment is stopped,
and the capsule is quenched to prevent the minerals/ liquids from back-reacting or reequilibrating at lower P and T. The capsule is opened, and the phases are identified
and analyzed chemically. Glass in the capsule would represent the melt that was in
equilibrium with whatever minerals coexisting with it. Each experiment would
therefore yield one point on a phase diagram. By plotting the results of many
experiments, we are finally able to draw the various phase boundaries (often with
some interpolation).
D- Chemical reactions
 Chemical reactions are either reversible or irreversible.
 Balancing chemical reactions of geological interest:
For chemical reactions involving aqueous species, follow these steps:
(i) conserve one element in the solid phases. In many cases, this is an element
that is relatively immobile. Depending on your choice of element to be
conserved, one could write several different reactions between the same
minerals.
(ii) balance all other elements. Balance for Si with SiO2 last.
(iii) balance the charge using H+ as a reactant or product
(iv) balance out the H+ by adding H2O as a reactant or product (last step after
balancing all other elements).
(v) double check your results: oxygen on both sides should now be equal.
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For reactions in which no aqueous species are allowed (i.e. all ions are conserved in
the solid phases; useful for metamorphic systems), use the following technique:
(i) Assume that the reaction is between the solid phases A + B = C + D
(ii) Balance the reaction between A (± B) and C conserving one element in the
solid phases, and using the same technique outlined above for reactions in
aqueous solutions
(iii) Now balance the reaction between B (± A) and D, conserving the same
element in the solid phases.
(iv) Multiply both reactions by appropriate factors to cancel out the unwanted
ionic aqueous species (such as H+, K+, … etc.)
(v) Add or subtract both reactions to get rid of all aqueous species.
Obviously, this technique can be pretty arduous, requiring many steps and iterations.
If that is the case, then use linear algebra to balance the equation. Here is the
technique used to setup your matrix:
(i) Break your phases up into their constituent oxides
(ii) # of phases = # of oxides + 1
(iii) Create your square matrix by setting up the columns as the minerals, and
the rows as the oxides. Reserve your last column for water. Leave out the
phase quartz, as that will be the matrix selected as your “product” for the
time being.
(iv) Invert your square matrix (can be done using an excel spreadsheet). Now
multiply the inverted matrix by the product matrix (the one you set up for
quartz). The resulting matrix gives you the stoichiometric coefficients you
need arranged in the same order as you arranged your original square
matrix.
E- Equilibrium
A system is said to be in equilibrium if:
(i) there is no observed change over time
(ii) for reversible equilibria: rate of forward reaction = rate of backward reaction
(iii) the chemical potential of all components in all phases is equal (see later)
(iv) the Gibbs free energy of the reaction = 0; energy of the system is minimum (see
later)
Types: Equilibrium can be one of three types: stable, metastable or unstable. A
stable equilibrium is one that will not change unless the variables of the system (e.g.
P, T, composition) are changed. A metastable equilibrium is one that appears
without change over time, but the system under such conditions does not have the
minimum energy, and will undergo changes as soon as small amounts of energy are
added to the system. An unstable equilibrium is one which appears not to change
over time, but actually does so slowly in an attempt to minimize the energy of the
system. In addition to these three types, there is the local equilibrium, where only
part of the system (i.e. some phases, but not all) is in equilibrium.
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Law of Mass Action:
"The rate of a reaction is directly proportional to the concentration of each reacting
substance"
Example: For some arbitrary reaction:
A+nBmC+D
where n and m are the stoichiometric reaction coefficients of C and B, respectively
K = [C]m [D]/[A] [B]n
where [C] is the concentration of “C”, … and so on.
 Holds for simple reactions, but as far as reaction rates are really concerned (i.e.
reaction kinetics), it is misleading!
 The value of "K" is reaction specific, i.e. it depends on how we write the reaction
(which are the products and which are the reactants).
 K depends on T & P!
Le Chatelier's Principle:
Any change in one of the variables that determine the state of a system in equilibrium
will result in a shift of the equilibrium in a direction that counteracts the change in
that variable. This is quite useful to make qualitative predictions.
The Solubility Product "Ksp":
Is a special kind of equilibrium constant when dealing with sparingly soluble salts
such as AgCl. In this case, the “concentration” of the solid is set as 1 (cf standard
states, and examples solved in class).
The Common Ion effect:
Is the decrease of the solubility of a salt due to the presence of one of its own ions in
solution. Note that the presence of ions different from those of the salt will generally
(but not always) make that salt more soluble.
F-Activity and activity coefficients
1- Activity for ionic species in solution:
Think of it as the effective concentration of a "component"!
ai = [i]
 At infinite dilution,  = 1
  is usually < 1, but in very concentrated solutions (high salinity brines) and some
minerals with non-ideal solid solutions (of course!),  may become > 1!!!
 a for ions and molecules is usually expressed in moles/liter, but molality, or
formality could be used (but be consistent!).
 “a” of pure solids and pure H2O = 1
 Reactions are assumed to take place at SATP, unless otherwise indicated.
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2- Fugacity: (For gas mixtures)
For gases, "effective concentrations" are expressed in terms of "fugacities" rather than
activities. Since concentrations for gases are expressed in terms of their partial pressures,
then:
fi = i.Pi
Where i is the fugacity coefficient (equivalent to  in the case of aqueous solutions), and
Pi is the partial pressure of component "i". For ideal gases, i = 1. The relationship
between fugacity and activity is given by:
ai = fi/fi°
where fi° is the fugacity of component “i” in the standard state.
3- Activities for crystalline (solid) solutions:
For minerals, the following relation holds:
ai = i Xi
where Xi is the mole fraction of component "i" (or endmember "i") in the mineral of
interest. For ideal crystalline solution, i.= 1.
Unlike electrolyte solutions, where activity coefficients are calculated largely on the basis
of empirical observations, calculation of activity coefficients in the case of minerals with
solid solutions requires a knowledge of basic thermodynamics.
G- Properties and Chemistry of Water and Aqueous Solutions
Structure of H2O:
 distorted tetrahedron (Fig. 5a)
  = 105°
 H - bonds hold molecules together
 Non-linear structure results in its dipolar nature (Fig. 5a)
 Interaction with ions: solvation (Fig. 5b)
The dielectric constant "":
 The dielectric constant is the reciprocal of the constant in Coulomb's law
expression for the force between electric charges contained in the substance.
F = C (q1. q2)/r2
i.e.   1/C
  = capacity of a condenser filled with H2O/ capacity of same condenser in
vacuum.
 Water has a very high dielectric constant which decreases with increasing T, but
increases with increasing P.
 The higher the value of , the greater the ability of water to dissolve compounds
(or to hold apart and prevent the reaction between, or the neutralization of,
solvated ions).
  varies with T (Fig. 7b).
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Dissolution and solvation (Fig. 5)
 Example: NaCl: Collapse of the open structure of H2O
 Solvated shells (Fig. 5b)
Phase diagram:
 Triple point (Fig. 6)
 Critical point: P = 220 bars, T = 374°C
Density:
 Variation with T and isochores (Fig. 7)
 Higest density at 4°C
The dissociation constant of water "Kw":
 Kw = [H+][OH-]
 At SATP, Kw = 10-14
 Effect of T on Kw (Fig. 8)
 Calculating pH from the value of Kw.
Conductivity:
 Conductance is the reciprocal of resistance.
 Conductance is expressed in “Seimens” (formerly known as “mho”).
 Conductivity is the “specific conductance” (measured between 2 opposite
faces of a 1 cm cube of material).
 Conductivty (S/cm) = 1/ resistivity (ohms – cm).
 Conductivity is an indirect measure of the TDS. Relationship is given by:
ppm TDS = conductivity (S/cm) x 0.66
H- Electrolyte Solutions & Ionic Strength
 The ionic strength of an aqueous solution is given by the expression:
I = 1/2  mizi2
where mi is the concentration (preferably molal) of species “i” in solution, and
z is the charge of this species.
 Obviously, the higher the salinity of a solution, the greater its ionic strength.
 The ionic strength is best obtained from a total water analysis, followed by the
application of the above formula. In the event that a full analysis is not
available, the value of I can be estimated from knowledge of the TDS (in ppm)
of your sample applying one of the following conventions:
I  2 x 10-5 x TDS for a NaCl solution
I  2.5 x 10-5 x TDS for average water
I  2.8 x 10-5 x TDS for carbonate waters
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“I” can also be estimated (in this case more accurately) from a measurement of
specific conductance (SpC or conductivity) in micromhos or microsiemens/ cm
using these formulations:
I  0.8 x 10-5 x SpC for a NaCl solution
I  1.7 x 10-5 x TDS for average water
I  1.9 x 10-5 x TDS for carbonate waters
 I described above is also known as the “stoichiometric ionic strength”, and can
be designated Is. Another type of ionic strength is known as the effective ionic
strength “Ie”, which applies a correction factor that accounts for ions backreaction with one another to form neutral complexes. The effective ionic
strength is therefore always less than the stoichiometric ionic strength, and is
typically 7% lower for average waters.
 If  is known, then the activity or effective calculation can be calculated
ai = s[is] = e[ie]
where [i] may be substituted for by mi if it is expressed in molal quantities.
Note that activities are independent of the type of ionic strength used.
 This non-ideal behavior therefore necessitates knowledge (or at least
estimation) of the activity coefficient  for each ion, which is no easy task.
However, in dilute solutions,  of strong (completely dissociated) electrolytes
will all be similar in solutions of the same ionic strength.
 The MacInnes Convention assumes that  for two monovalent ions of the same
salt are equal. Therefore, for NaCl Na = Cl. This helps us come up with
approximate values for  rather quickly. This works best if the two ions have
the same charges and sizes.
 Another approximation needed for a quick evaluation of  is to assume a 
value for the salt equal to the product of s of its individual ions raised to the
appropriate power. For example, for K2SO4, K2SO4 = [(2k)( SO4)]1/3. This is
often referred to as the mean ion activity approach.
Calculation of  for ions in electrolyte solutions of different ionic strengths:
Although the following methods are still empirical, they are generally better than the
mean ionic strength method and the MacInnes convention. The following methods
take into consideration the ionic strength of the solution as well as the factors that
affect !!!
Factors affecting the value of :
(i) Presence of other ions
(ii) The concentrations of the other interfering ions
(iii) The charges of the interfering ions (Fig. 9).
(iv) The charge of the ion of interest (z):  is larger for higher z.
(v) The size of the ion of interest (r)
(vi) The nature of the solvent, particularly its density and dielectric constant (a
measure of the polarity of its molecules; e.g. for H2O, it is a measure of the
ability of H2O dipoles to rearrange themselves in an applied electric field).
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Methods of calculating 
(i) The Debye - Huckel Theory: Applied to dilute solutions with I < 5 . 10-3
The original DH formulation (also known as the DH limiting law) is
-log  = Az2I1/2 …………………………………….(1)
where A = 0.5085 for H2O at 25°C.
(ii) For I > 5 . 10-3 but < 0.1
Use the extended or modified DH equation:
-log  = Az2I1/2/(1+aBI1/2)
…………………..(2)
where “a” is the radius of the hydrated ion (again determined empirically!), and A &
“B” are constants that depend on P, T, , and . At SATP, A = 0.5092, and B =
0.3283. Figure 9 shows  as a function of I for different ions with different values of
“a”.
Note that for many ions, aB ~ 1. The equation therefore can be rewritten as
-log  = Az2I1/2/(1+I1/2)
………………… (3)
(known as the Guntelberg equation)
(iii) Davies Equation: for solutions with I of up to 0.5
-log  = {Az2I1/2/(1+aBI1/2)} + bI ……………………(4)
where “b” is a constant specific to the individual ion, but which is commonly set at
0.3. This formulation does not have a term for ion size, and therefore gives  values
that are the same for all ions with the same z.
(iv) The Truesdell – Jones equation
This uses the extended DH equation plus an add on term specific for each ion, which
attempts to correct for the size of each ion empirically.
(iv) The Specific ion interaction theory (SIT): for solutions with I of up to 3.5
This equation is theoretically capable of greater accuracy than the Davies approach,
and is given by:
log (i) = -z2D +  (i,j,I) m(j)
Where D is the Debye Huckel term (D = 0.5091 I1/2/(1 + 1.5 I1/2)), (i,j,I) is an
interaction parameter between ions I and j for a given ionic strength I, and m(j) is the
molality of the major ion of opposite charge to ion i.
Figure 10 shows  as a function of I using some of the equations listed above,
whereas Fig. 11 shows a comparison between the different results obtained by these
methods.
For more concentrated solutions, see equations in Drever (1997) p. 31 – 37, and
your text p. 138 (Pitzer model).
Activity coefficients for molecular species: salting in and salting out effects (Fig.
12).